Wave vector
Encyclopedia
In physics
, a wave vector (also spelled wavevector) is a vector which helps describe a wave
. Like any vector, it has a magnitude and direction, both of which are important: Its magnitude is either the wavenumber
or angular wavenumber of the wave (inversely proportional to the wavelength
), and its direction is ordinarily the direction of wave propagation
(but not always, see below).
The wave vector can also be defined as a four-vector
in the context of special relativity
.
in their magnitudes. In general, one definition is preferred in physics
and related fields, while the other definition is preferred in crystallography
and related fields. For this article, they will be called the "physics definition" and the "crystallography definition", respectively.
where:
This wave travels in the +x direction with speed (more specifically, phase velocity
)
.
This formula is generalized to three dimensions by:
where:
This formula describes a plane wave
. The magnitude of the wave vector is the angular wavenumber as in the one-dimensional case above:
. The direction of the wave vector is ordinarily the direction that the plane wave is traveling, but it can differ slightly in an anisotropic medium
. (See below).
, the same waves are described using slightly different equations. In one and three dimensions respectively:
The differences are:
The direction of k is discussed below.
". The direction of wave propagation is the direction of a wave's energy flow, and the direction that a small wave packet
will move. For light waves, this is the direction of the Poynting vector
.
In an isotropic medium
such as air, any gas, any liquid, or some solids (such as glass
), the direction of the wavevector is exactly the same as the direction of wave propagation.
However, when a wave travels through an anisotropic medium
, such as light waves through an asymmetric crystal
or sound waves through a sedimentary rock
, the wave vector may not point exactly in the direction of wave propagation.
The general direction of the wave vector, which is true in both isotropic and anisotropic media, is that it points in the normal direction
to the surfaces of constant phase, also called wave fronts.
which, when written out explicitly in its contravariant and covariant forms is
The null character of the wave 4-vector gives a relation between the frequency and the magnitude of the spatial part of the wave 4-vector:
of the wave vector is one way to derive the relativistic Doppler effect
. The Lorentz matrix is defined as
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, a wave vector (also spelled wavevector) is a vector which helps describe a wave
Wave
In physics, a wave is a disturbance that travels through space and time, accompanied by the transfer of energy.Waves travel and the wave motion transfers energy from one point to another, often with no permanent displacement of the particles of the medium—that is, with little or no associated mass...
. Like any vector, it has a magnitude and direction, both of which are important: Its magnitude is either the wavenumber
Wavenumber
In the physical sciences, the wavenumber is a property of a wave, its spatial frequency, that is proportional to the reciprocal of the wavelength. It is also the magnitude of the wave vector...
or angular wavenumber of the wave (inversely proportional to the wavelength
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
), and its direction is ordinarily the direction of wave propagation
Wave propagation
Wave propagation is any of the ways in which waves travel.With respect to the direction of the oscillation relative to the propagation direction, we can distinguish between longitudinal wave and transverse waves....
(but not always, see below).
The wave vector can also be defined as a four-vector
Four-vector
In the theory of relativity, a four-vector is a vector in a four-dimensional real vector space, called Minkowski space. It differs from a vector in that it can be transformed by Lorentz transformations. The usage of the four-vector name tacitly assumes that its components refer to a standard basis...
in the context of special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
.
Definitions
Unfortunately, there are two common definitions of wave vector which differ by a factor of in their magnitudes. In general, one definition is preferred in physicsPhysics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
and related fields, while the other definition is preferred in crystallography
Crystallography
Crystallography is the experimental science of the arrangement of atoms in solids. The word "crystallography" derives from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and grapho = write.Before the development of...
and related fields. For this article, they will be called the "physics definition" and the "crystallography definition", respectively.
Physics definition
A perfect one-dimensional traveling wave follows the equation:where:
- x is position,
- t is time,
(a function of x and t) is the disturbance describing the wave (for example, for an ocean wave, 
would be the excess height of the water, or for a sound wave, 
would be the excess air pressure).- A is the amplitudeAmplitudeAmplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...
of the wave (the peak magnitude of the oscillation), 
is a "phase offset" describing how two waves can be out of sync with each other,
is the angular frequencyAngular frequencyIn physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
of the wave, related to how quickly it oscillates at a given point,
is the wavenumberWavenumberIn the physical sciences, the wavenumber is a property of a wave, its spatial frequency, that is proportional to the reciprocal of the wavelength. It is also the magnitude of the wave vector...
(more specifically called angular wavenumber) of the wave, related to the wavelengthWavelengthIn physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
by the equation
.
This wave travels in the +x direction with speed (more specifically, phase velocity
Phase velocity
The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the speed at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave will appear to travel at the phase velocity...
)
.This formula is generalized to three dimensions by:
where:
- r is the position vector in three-dimensional space,
is the vector dot productDot productIn mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...
.- k is the wave vector.
This formula describes a plane wave
Plane wave
In the physics of wave propagation, a plane wave is a constant-frequency wave whose wavefronts are infinite parallel planes of constant peak-to-peak amplitude normal to the phase velocity vector....
. The magnitude of the wave vector is the angular wavenumber as in the one-dimensional case above:
. The direction of the wave vector is ordinarily the direction that the plane wave is traveling, but it can differ slightly in an anisotropic mediumAnisotropy
Anisotropy is the property of being directionally dependent, as opposed to isotropy, which implies identical properties in all directions. It can be defined as a difference, when measured along different axes, in a material's physical or mechanical properties An example of anisotropy is the light...
. (See below).
Crystallography definition
In crystallographyCrystallography
Crystallography is the experimental science of the arrangement of atoms in solids. The word "crystallography" derives from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and grapho = write.Before the development of...
, the same waves are described using slightly different equations. In one and three dimensions respectively:
The differences are:
- The frequency
instead of angular frequency 
is used. They are related by 
. This substitution is not important for this article, but reflects common practice in crystallography. - The wavenumber k and wave vector k are defined in a different way. Here,
, while in the physics definition above, 
.
The direction of k is discussed below.
Direction of the wave vector
The direction in which the wave vector points must be distinguished from the "direction of wave propagationWave propagation
Wave propagation is any of the ways in which waves travel.With respect to the direction of the oscillation relative to the propagation direction, we can distinguish between longitudinal wave and transverse waves....
". The direction of wave propagation is the direction of a wave's energy flow, and the direction that a small wave packet
Wave packet
In physics, a wave packet is a short "burst" or "envelope" of wave action that travels as a unit. A wave packet can be analyzed into, or can be synthesized from, an infinite set of component sinusoidal waves of different wavenumbers, with phases and amplitudes such that they interfere...
will move. For light waves, this is the direction of the Poynting vector
Poynting vector
In physics, the Poynting vector can be thought of as representing the directional energy flux density of an electromagnetic field. It is named after its inventor John Henry Poynting. Oliver Heaviside and Nikolay Umov independently co-invented the Poynting vector...
.
In an isotropic medium
Isotropy
Isotropy is uniformity in all orientations; it is derived from the Greek iso and tropos . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix an, hence anisotropy. Anisotropy is also used to describe situations where properties vary...
such as air, any gas, any liquid, or some solids (such as glass
Glass
Glass is an amorphous solid material. Glasses are typically brittle and optically transparent.The most familiar type of glass, used for centuries in windows and drinking vessels, is soda-lime glass, composed of about 75% silica plus Na2O, CaO, and several minor additives...
), the direction of the wavevector is exactly the same as the direction of wave propagation.
However, when a wave travels through an anisotropic medium
Anisotropy
Anisotropy is the property of being directionally dependent, as opposed to isotropy, which implies identical properties in all directions. It can be defined as a difference, when measured along different axes, in a material's physical or mechanical properties An example of anisotropy is the light...
, such as light waves through an asymmetric crystal
Crystal optics
Crystal optics is the branch of optics that describes the behaviour of light in anisotropic media, that is, media in which light behaves differently depending on which direction the light is propagating. The index of refraction depends on both composition and crystal structure and can be...
or sound waves through a sedimentary rock
Sedimentary rock
Sedimentary rock are types of rock that are formed by the deposition of material at the Earth's surface and within bodies of water. Sedimentation is the collective name for processes that cause mineral and/or organic particles to settle and accumulate or minerals to precipitate from a solution....
, the wave vector may not point exactly in the direction of wave propagation.
The general direction of the wave vector, which is true in both isotropic and anisotropic media, is that it points in the normal direction
Surface normal
A surface normal, or simply normal, to a flat surface is a vector that is perpendicular to that surface. A normal to a non-flat surface at a point P on the surface is a vector perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a...
to the surfaces of constant phase, also called wave fronts.
In special relativity
A beam of coherent, monochromatic light can be characterized by the (null) wave 4-vector
which, when written out explicitly in its contravariant and covariant forms is
-
and
The null character of the wave 4-vector gives a relation between the frequency and the magnitude of the spatial part of the wave 4-vector:
Lorentz transformation
Taking the Lorentz transformationLorentz transformation
In physics, the Lorentz transformation or Lorentz-Fitzgerald transformation describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frames of reference. It is named after the Dutch physicist Hendrik...
of the wave vector is one way to derive the relativistic Doppler effect
Relativistic Doppler effect
The relativistic Doppler effect is the change in frequency of light, caused by the relative motion of the source and the observer , when taking into account effects described by the special theory of relativity.The relativistic Doppler effect is different from the non-relativistic Doppler effect...
. The Lorentz matrix is defined as
-
In the situation where light is being emitted by a fast moving source and one would like to know the frequency of light detected in an earth (lab) frame, we would apply the Lorentz transformation as follows. Note that the source is in a frame Ss and earth is in the observing frame, Sobs.
Applying the lorentz transformation to the wave vector
and choosing just to look at the
component results in
-
- {|
|
|
|-
|
|
where
is the direction cosine of 
wrt 
|}
So-
- {|cellpadding="2" style="border:2px solid #ccccff"
|
|}
Source moving away
As an example, to apply this to a situation where the source is moving directly away from the observer (
), this becomes:
Source moving towards
To apply this to a situation where the source is moving straight towards the observer (
), this becomes:
-
